How can I restore the rights to my sister-in-law from whom I stole, and how can I atone for my false oath after denying the theft and having the Qur’an placed before me to swear upon?

Repentance to Allah is obligatory for what was taken unjustly, through remorse, cessation, and determination, along with returning the right to its owner without informing her of what you did. The false oath (Yamin Ghamus) is one of the major sins, and repentance from it is obligatory. However, according to the majority of scholars, it does not require an expiation for an oath.